Verify that the area of the triangle with vertices (2, 3), (5, 7) and (-3 -1) remains invariant under the translation of axes when the origin is shifted to the point (-1, 3).
Given points (2, 3), (5, 7), and (-3, -1).
To show: The area of a triangle is invariant to shifting of origin.
The area of triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is
= 
[x1(y2 – y3) + x2(y3 -y1) + x3(y1 – y2)]
Hence, the area of given triangle = 
[2(7+1) + 5(-1-3) – 3(3-7)]
= 
[16 – 20 + 12]
= 
[8]
= 4
Origin shifted to point (-1, 3), the new coordinates of the triangle are (3, 0), (6, 4), and (-2, -4) obtained from subtracting a point (-1, 3).
Hence, the new area of triangle = 
[3(4-(-4)) + 6(-4-0) – 2(0-4)]
= 
[24-24+8]
= 
[8]
= 4
Since the area of the triangle before and after the translation after shifting of origin remains same, i.e. 4. Therefore we can say that the area of a triangle is invariant to shifting of origin.